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We already have studied four of the five type classes in the Prelude that can be used for data structure manipulation: Functor, Applicative, Monad and Foldable. The fifth one is Traversable [1]. To traverse means to walk across, and that is exactly what Traversable generalises: walking across a structure, collecting results at each stop.

Functors made for walking[edit | edit source]

If traversing means walking across, though, we have been performing traversals for a long time already. Consider the following plausible Functor and Foldable instances for the Tree from Other data structures:

data Tree a = Leaf a | Branch (Tree a) (Tree a)
            deriving (Eq, Show)

instance Functor Tree where
    fmap f (Leaf x)       = Leaf (f x)
    fmap f (Branch tx ty) = Branch (f <$> tx) (f <$> ty)

instance Foldable Tree where
    foldMap f (Leaf x)       = f x
    foldMap f (Branch tx ty) = foldMap f tx <> foldMap f ty

fmap f walks across the tree, applies f to each element and collects the results by rebuilding the tree. Similarly, foldMap f walks across the tree, applies f to each element and collects the results by combining them with mappend. Functor and Foldable, however, are not enough to express all useful ways of traversing Tree. For instance, suppose we have the following Maybe-encoded test for negative numbers...

deleteIfNegative :: (Num a, Ord a) => a -> Maybe a
deleteIfNegative x = if x < 0 then Nothing else Just x

... and we want to use it to implement...

rejectWithNegatives :: (Num a, Ord a) => Tree a -> Maybe (Tree a)

... which gives back the original tree wrapped in Just if there are no negative elements in it, and Nothing otherwise. Neither Foldable nor Functor on their own would help. Using Foldable would replace the structure of the original tree with that of whatever Monoid we pick for folding, and there is no way to encode the structure of Tree into a Monoid [2]. As for Functor, fmap might be attractive at first...

GHCi> let testTree = Branch (Branch (Leaf 5) (Leaf (-3))) (Leaf 7)
GHCi> fmap deleteIfNegative testTree
Branch (Branch (Leaf (Just 5)) (Leaf Nothing)) (Leaf (Just 7))

... but then we would need a way to turn a Tree of Maybe into Maybe a Tree. If you squint hard enough, that looks somewhat like a fold. Instead, however, of merely combining the values with mappend (which destroys the tree structure), we need to combine the Maybe contexts of the values and recreate the tree structure within the combined context. Fortunately, there is a type class that is essentially about combining Functor contexts: Applicative [3]. Applicative, in turn, leads us to the class we need: Traversable.

instance Traversable [] where
    -- sequenceA :: Applicative f => Tree (f a) -> f (Tree a)
    sequenceA (Leaf x)       = Leaf <$> x
    sequenceA (Branch tx ty) = Branch <$> sequenceA tx <*> sequenceA ty

Traversable is to Applicative contexts what Foldable is to Monoid values. From that point of view, sequenceA is analogous to fold − it creates an applicative summary of the contexts within a structure, and then rebuilds the structure in the new context. sequenceA is the function we were looking for:

GHCi> let rejectWithNegatives = sequenceA . fmap deleteIfNegative
GHCi> :t rejectWithNegatives 
rejectWithNegatives
  :: (Num a, Ord a, Traversable t) => t a -> Maybe (t a)
GHCi> rejectWithNegatives testTree
Nothing
GHCi> let testTree' = Branch (Branch (Leaf 1) (Leaf 2)) (Leaf 3)
GHCi> rejectWithNegatives testTree'
Just (Branch (Branch (Leaf 1) (Leaf 2)) (Leaf 3))

These are the methods of Traversable:

class (Functor t, Foldable t) => Traversable t where
    traverse  :: Applicative f => (a -> f b) -> t a -> f (t b)
    sequenceA :: Applicative f => t (f a) -> f (t a)

    -- These methods have default definitions.
    -- They are merely specialised versions of the other two.
    mapM      :: Monad m => (a -> m b) -> t a -> m (t b)
    sequence  :: Monad m => t (m a) -> m (t a)

If sequenceA is analogous to fold, traverse is analogous to foldMap. They can be defined in terms of each other, and therefore a minimal implementation of Traversable just needs to supply one of them:

traverse f = sequenceA . fmap f
sequenceA = traverse id

Rewriting the list instance using traverse makes the parallels with Functor and Foldable obvious:

instance Traversable Tree where
    traverse f (Leaf x)       = Leaf <$> f x
    traverse f (Branch tx ty) = Branch <$> traverse f tx <*> traverse f ty

In general, it is better to write traverse when implementing Traversable, as the default definition of traverse performs, in principle, two runs across the structure (one for fmap and another for sequenceA).

We can cleanly define rejectWithNegatives directly in terms of traverse:

rejectWithNegatives :: (Num a, Ord a, Traversable t) => t a -> Maybe (t a)
rejectWithNegatives = traverse deleteIfNegative
Exercises
  1. Write a Traversable instance for the standard list type. Do not use foldMap or foldr.

Interpretations of Traversable[edit | edit source]

Traversable structures can be walked over using the applicative functor of your choice. The type of traverse...

traverse :: (Applicative f, Traversable t) => (a -> f b) -> t a -> f (t b)

... resembles that of mapping functions we have seen in other classes. Rather than using its function argument to insert functorial contexts under the original structure (as might be done with fmap) or to modify the structure itself (as (>>=) does), traverse adds an extra layer of context on the top of the structure. Said in another way, traverse allows for effectful traversals − traversals which produce an overall effect (i.e. the new outer layer of context).

If the structure below the new layer is recoverable at all, it will match the original structure (the values might have changed, of course). Here is an example involving nested lists:

GHCi> traverse (\x -> [0..x]) [0..3]
[[0,0,0,0],[0,0,0,1],[0,0,0,2],[0,0,0,3],[0,0,1,0],[0,0,1,1]
,[0,0,1,2],[0,0,1,3],[0,0,2,0],[0,0,2,1],[0,0,2,2],[0,0,2,3]
,[0,1,0,0],[0,1,0,1],[0,1,0,2],[0,1,0,3],[0,1,1,0],[0,1,1,1]
,[0,1,1,2],[0,1,1,3],[0,1,2,0],[0,1,2,1],[0,1,2,2],[0,1,2,3]
]

The inner lists retain the structure the original list − all of them have four elements. The outer list is the new layer, corresponding to the introduction of nondeterminism through allowing each element to vary from zero to its (original) value.

We can also understand Traversable by focusing on sequenceA and how it distributes context.

GHCi> sequenceA [[1,2,3,4],[5,6,7]]
[[1,5],[1,6],[1,7],[2,5],[2,6],[2,7]
,[3,5],[3,6],[3,7],[4,5],[4,6],[4,7]
]

In this example, sequenceA can be seen distributing the old outer structure into the new outer structure, and so the new inner lists have two elements, just like the old outer list. The new outer structure is a list of twelve elements, which is exactly what you would expect from combining with (<*>) one list of four elements with another of three elements. One interesting aspect of the distribution perspective is how it helps making sense of why certain functors cannot possibly have instances of Traversable (how would one distribute an IO action? Or a function?).

Exercises

Having the applicative functors chapter fresh in memory can help with the following exercises.

  1. Consider a representation of matrices as nested lists, with the inner lists being the rows. Use Traversable to implement
    transpose :: [[a]] -> [[a]]
    which transposes a matrix (i.e. changes columns into rows and vice-versa). For the purposes of this exercise, we don't care about how fake "matrices" with rows of different sizes are handled.
  2. Explain what traverse mappend does.
  3. Time for a round of Spot The Applicative Functor. Consider:
    mapAccumL :: Traversable t =>
    (a -> b -> (a, c)) -> a -> t b -> (a, t c)

    Does its type remind you of anything? Use the appropriate Applicative to implement it with Traversable. As further guidance, here is the description of mapAccumL in the Data.Traversable documentation:

    The mapAccumL function behaves like a combination of fmap and foldl; it applies a function to each element of a structure, passing an accumulating parameter from left to right, and returning a final value of this accumulator together with the new structure.

The Traversable laws[edit | edit source]

Sensible instances of Traversable have a set of laws to follow. There are the following two laws:

traverse Identity = Identity -- identity
traverse (Compose . fmap g . f) = Compose . fmap (traverse g) . traverse f -- composition

Plus a bonus law, which is guaranteed to hold:

-- If t is an applicative homomorphism, then
t . traverse f = traverse (t . f) -- naturality

Those laws are not exactly self-explanatory, so let's have a closer look at them. Starting from the last one: an applicative homomorphism is a function which preserves the Applicative operations, so that:

-- Given a choice of f and g, and for any a,
t :: (Applicative f, Applicative g) => f a -> g a

t (pure x) = pure x
t (x <*> y) = t x <*> t y

Note that not only this definition is analogous to the one of monoid homomorphisms which we have seen earlier on but also that the naturality law mirrors exactly the property about foldMap and monoid homomorphisms seen in the chapter about Foldable.

The identity law involves Identity, the dummy functor:

newtype Identity a = Identity { runIdentity :: a }

instance Functor Identity where
    fmap f (Identity x) = Identity (f x)

instance Applicative Identity where
    pure x = Identity x
    Identity f <*> Identity x = Identity (f x)

The law says that all traversing with the Identity constructor does is wrapping the structure with Identity, which amounts to doing nothing (as the original structure can be trivially recovered with runIdentity). The Identity constructor is thus the identity traversal, which is very reasonable indeed.

The composition law, in turn, is stated in terms of the Compose functor:

newtype Compose f g a = Compose { getCompose :: f (g a) }

instance (Functor f, Functor g) => Functor (Compose f g) where
    fmap f (Compose x) = Compose (fmap (fmap f) x)

instance (Applicative f, Applicative g) => Applicative (Compose f g) where
    pure x = Compose (pure (pure x))
    Compose f <*> Compose x = Compose ((<*>) <$> f <*> x)

Compose performs composition of functors. Composing two Functors results in a Functor, and composing two Applicatives results in an Applicative [4]. The instances are the obvious ones, threading the methods one further functorial layer down.

The composition law states that it doesn't matter whether we perform two traversals separately (right side of the equation) or compose them in order to walk across the structure only once (left side). It is analogous, for instance, to the second functor law. The fmaps are needed because the second traversal (or the second part of the traversal, for the left side of the equation) happens below the layer of structure added by the first (part). Compose is needed so that the composed traversal is applied to the correct layer.

Identity and Compose are available from Data.Functor.Identity and Data.Functor.Compose respectively.

The laws can also be formulated in terms of sequenceA:

sequenceA . fmap Identity = Identity -- identity
sequenceA . fmap Compose = Compose . fmap sequenceA . sequenceA -- composition
-- For any applicative homomorphism t:
t . sequenceA = sequenceA . fmap t -- naturality

Though it's not immediately obvious, several desirable characteristics of traversals follow from the laws, including [5]:

  • Traversals do not skip elements.
  • Traversals do not visit elements more than once.
  • traverse pure = pure
  • Traversals cannot modify the original structure (it is either preserved or fully destroyed).

Recovering fmap and foldMap[edit | edit source]

We still have not justified the Functor and Foldable class constraints of Traversable. The reason for them is very simple: as long as the Traversable instance follows the laws traverse is enough to implement both fmap and foldMap. For fmap, all we need is to use Identity to make a traversal out of an arbitrary function:

fmap f = runIdentity . traverse (Identity . f)

To recover foldMap, we need to introduce a third utility functor: Const from Control.Applicative:

newtype Const a b = Const { getConst :: a }

instance Functor (Const a) where
    fmap _ (Const x) = Const x

Const is a constant functor. A value of type Const a b does not actually contain a b value. Rather, it holds an a value which is unaffected by fmap. For our current purposes, the truly interesting instance is the Applicative one

instance Monoid a => Applicative (Const a) where
    pure _ = Const mempty
    Const x <*> Const y = Const (x `mappend` y)

(<*>) simply combines the values in each context with mappend [6]. We can exploit that to make a traversal out of any Monoid m => a -> m function that we might pass to foldMap. Thanks to the instance above, the traversal then becomes a fold:

foldMap f = getConst . traverse (Const . f)

We have just recovered from traverse two functions which on the surface appear to be entirely different, and all we had to do was pick two different functors. That is a taste of how powerful an abstraction functors are [7].

Non-equivalence of Traversable and Foldable[edit | edit source]

Above, we discovered that foldMap can be implemented in terms of traverse. However, is the converse true? That is, can traverse be implemented in terms of foldMap?

Let's take another look at the Foldable and Traversable instances for the Tree type considered previously.

instance Foldable Tree where
    foldMap f (Leaf x)       = f x
    foldMap f (Branch tx ty) = foldMap f tx <> foldMap f ty

instance Traversable Tree where
    traverse f (Leaf x)       = Leaf <$> f x
    traverse f (Branch tx ty) = Branch <$> traverse f tx <*> traverse f ty

While the definitions are very similar, we see that traverse rebuilds the tree structure (as required by the Traversable laws) within the resulting Applicative context, while foldMap simply combines the resulting monoid values using mappend. As we saw above, Const is an Applicative functor that mimics the behavior of a monoid, and this is what allowed us to define foldMap in terms of traverse. Therefore, implementing traverse in terms of foldMap boils down to finding a monoid that mimics the behavior of Applicative.

Rewriting the above instances with each case grouped together makes things a bit more clear:

foldMap  f (Leaf x) = f               x
traverse f (Leaf x) = (fmap Leaf . f) x

foldMap  f (Branch tx ty) = mappend       (foldMap f tx)  (foldMap f ty)
traverse f (Branch tx ty) = liftA2 Branch (traverse f tx) (traverse f ty)

The obvious attempt would be to let traverse f = foldMap (fmap Leaf . f) and define a monoid such that mappend = liftA2 Branch. However, this will not work! Because Branch is not associative, liftA2 Branch is also not associative and therefore is not a valid definition for mappend. In addition, there is no reasonable definition for mempty in this case, since there is no empty Tree. It turns out that implementing traverse in terms foldMap alone is simply impossible for Tree, and hence cannot be done in general.

We've shown that implementing traverse in terms of foldMap alone is impossible for Tree, but what about other Traversable structures? The difficulty with Tree is that traverse needs to rebuild the structure by mapping each of its constructors over the functor contexts. Applicative provides the flexibility to make this work for constructors that have any number of arguments, and it does not impose any restraints on the functions (such as associativity). Foldable, on the other hand, does not support constructing types with anything more than an empty constructor (e.g. [] or Nothing) via mempty, a singleton constructor (e.g. (: []) or Just) via the f in foldMap f, and a binary associative constructor (e.g. (++)) via mappend such that mempty is a left and right identity. The most complex type that can be constructed by Foldable is therefore a list, and indeed a list traversal can be implemented as a fold using an appropriate Monoid instance:

newtype FList f a = FList { unFList :: f [a] }

instance Applicative f => Monoid (FList f a) where
    mempty = FList $ pure []
    FList x `mappend` FList y = FList $ liftA2 (++) x y

traverseList :: Applicative f => (a -> f b) -> [a] -> f [b]
traverseList f = unFList . foldMap (FList . fmap (: []) . f)

-- Equivalently, we can avoid the Monoid instance and use foldr
-- directly:
traverseList f = foldr (liftA2 (:) . f) (pure [])


Notes

  1. Strictly speaking, we should refer to the five classes in the GHC Prelude, as Applicative, Foldable and Traversable aren't officially part of the Prelude yet according to the Haskell Report. It is just a matter of time for them to be included, though.
  2. This is primarily because the constructor B is not associative. See the section on Non-equivalence of Traversable and Foldable.
  3. The monoidal presentation of Applicative makes that very clear.
  4. Remarkably, however, composing two Monads does not necessarily result in a Monad.
  5. For technical details, check the papers cited by the Data.Traversable documentation.
  6. This is a great illustration of how Applicative combines contexts monoidally. If we remove the values within the context, the applicative laws in monoidal presentation match the monoid laws exactly.
  7. A prime example, and one of clear practical relevance at that, is that great ode to functors, the lens library.